// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2009-2010 Benoit Jacob <jacob.benoit.1@gmail.com>
// Copyright (C) 2013-2014 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#ifndef EIGEN_JACOBISVD_H
#define EIGEN_JACOBISVD_H

namespace Eigen {

namespace internal {
    // forward declaration (needed by ICC)
    // the empty body is required by MSVC
    template <typename MatrixType, int QRPreconditioner, bool IsComplex = NumTraits<typename MatrixType::Scalar>::IsComplex>
    struct svd_precondition_2x2_block_to_be_real
    {
    };

    /*** QR preconditioners (R-SVD)
 ***
 *** Their role is to reduce the problem of computing the SVD to the case of a square matrix.
 *** This approach, known as R-SVD, is an optimization for rectangular-enough matrices, and is a requirement for
 *** JacobiSVD which by itself is only able to work on square matrices.
 ***/

    enum
    {
        PreconditionIfMoreColsThanRows,
        PreconditionIfMoreRowsThanCols
    };

    template <typename MatrixType, int QRPreconditioner, int Case> struct qr_preconditioner_should_do_anything
    {
        enum
        {
            a = MatrixType::RowsAtCompileTime != Dynamic && MatrixType::ColsAtCompileTime != Dynamic &&
                MatrixType::ColsAtCompileTime <= MatrixType::RowsAtCompileTime,
            b = MatrixType::RowsAtCompileTime != Dynamic && MatrixType::ColsAtCompileTime != Dynamic &&
                MatrixType::RowsAtCompileTime <= MatrixType::ColsAtCompileTime,
            ret = !((QRPreconditioner == NoQRPreconditioner) || (Case == PreconditionIfMoreColsThanRows && bool(a)) ||
                    (Case == PreconditionIfMoreRowsThanCols && bool(b)))
        };
    };

    template <typename MatrixType,
              int QRPreconditioner,
              int Case,
              bool DoAnything = qr_preconditioner_should_do_anything<MatrixType, QRPreconditioner, Case>::ret>
    struct qr_preconditioner_impl
    {
    };

    template <typename MatrixType, int QRPreconditioner, int Case> class qr_preconditioner_impl<MatrixType, QRPreconditioner, Case, false>
    {
    public:
        void allocate(const JacobiSVD<MatrixType, QRPreconditioner>&) {}
        bool run(JacobiSVD<MatrixType, QRPreconditioner>&, const MatrixType&) { return false; }
    };

    /*** preconditioner using FullPivHouseholderQR ***/

    template <typename MatrixType> class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
    {
    public:
        typedef typename MatrixType::Scalar Scalar;
        enum
        {
            RowsAtCompileTime = MatrixType::RowsAtCompileTime,
            MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime
        };
        typedef Matrix<Scalar, 1, RowsAtCompileTime, RowMajor, 1, MaxRowsAtCompileTime> WorkspaceType;

        void allocate(const JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd)
        {
            if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
            {
                m_qr.~QRType();
                ::new (&m_qr) QRType(svd.rows(), svd.cols());
            }
            if (svd.m_computeFullU)
                m_workspace.resize(svd.rows());
        }

        bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
        {
            if (matrix.rows() > matrix.cols())
            {
                m_qr.compute(matrix);
                svd.m_workMatrix = m_qr.matrixQR().block(0, 0, matrix.cols(), matrix.cols()).template triangularView<Upper>();
                if (svd.m_computeFullU)
                    m_qr.matrixQ().evalTo(svd.m_matrixU, m_workspace);
                if (svd.computeV())
                    svd.m_matrixV = m_qr.colsPermutation();
                return true;
            }
            return false;
        }

    private:
        typedef FullPivHouseholderQR<MatrixType> QRType;
        QRType m_qr;
        WorkspaceType m_workspace;
    };

    template <typename MatrixType> class qr_preconditioner_impl<MatrixType, FullPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
    {
    public:
        typedef typename MatrixType::Scalar Scalar;
        enum
        {
            RowsAtCompileTime = MatrixType::RowsAtCompileTime,
            ColsAtCompileTime = MatrixType::ColsAtCompileTime,
            MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
            MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
            Options = MatrixType::Options
        };

        typedef
            typename internal::make_proper_matrix_type<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime>::type
                TransposeTypeWithSameStorageOrder;

        void allocate(const JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd)
        {
            if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
            {
                m_qr.~QRType();
                ::new (&m_qr) QRType(svd.cols(), svd.rows());
            }
            m_adjoint.resize(svd.cols(), svd.rows());
            if (svd.m_computeFullV)
                m_workspace.resize(svd.cols());
        }

        bool run(JacobiSVD<MatrixType, FullPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
        {
            if (matrix.cols() > matrix.rows())
            {
                m_adjoint = matrix.adjoint();
                m_qr.compute(m_adjoint);
                svd.m_workMatrix = m_qr.matrixQR().block(0, 0, matrix.rows(), matrix.rows()).template triangularView<Upper>().adjoint();
                if (svd.m_computeFullV)
                    m_qr.matrixQ().evalTo(svd.m_matrixV, m_workspace);
                if (svd.computeU())
                    svd.m_matrixU = m_qr.colsPermutation();
                return true;
            }
            else
                return false;
        }

    private:
        typedef FullPivHouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
        QRType m_qr;
        TransposeTypeWithSameStorageOrder m_adjoint;
        typename internal::plain_row_type<MatrixType>::type m_workspace;
    };

    /*** preconditioner using ColPivHouseholderQR ***/

    template <typename MatrixType> class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
    {
    public:
        void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd)
        {
            if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
            {
                m_qr.~QRType();
                ::new (&m_qr) QRType(svd.rows(), svd.cols());
            }
            if (svd.m_computeFullU)
                m_workspace.resize(svd.rows());
            else if (svd.m_computeThinU)
                m_workspace.resize(svd.cols());
        }

        bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
        {
            if (matrix.rows() > matrix.cols())
            {
                m_qr.compute(matrix);
                svd.m_workMatrix = m_qr.matrixQR().block(0, 0, matrix.cols(), matrix.cols()).template triangularView<Upper>();
                if (svd.m_computeFullU)
                    m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace);
                else if (svd.m_computeThinU)
                {
                    svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols());
                    m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace);
                }
                if (svd.computeV())
                    svd.m_matrixV = m_qr.colsPermutation();
                return true;
            }
            return false;
        }

    private:
        typedef ColPivHouseholderQR<MatrixType> QRType;
        QRType m_qr;
        typename internal::plain_col_type<MatrixType>::type m_workspace;
    };

    template <typename MatrixType> class qr_preconditioner_impl<MatrixType, ColPivHouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
    {
    public:
        typedef typename MatrixType::Scalar Scalar;
        enum
        {
            RowsAtCompileTime = MatrixType::RowsAtCompileTime,
            ColsAtCompileTime = MatrixType::ColsAtCompileTime,
            MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
            MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
            Options = MatrixType::Options
        };

        typedef
            typename internal::make_proper_matrix_type<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime>::type
                TransposeTypeWithSameStorageOrder;

        void allocate(const JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd)
        {
            if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
            {
                m_qr.~QRType();
                ::new (&m_qr) QRType(svd.cols(), svd.rows());
            }
            if (svd.m_computeFullV)
                m_workspace.resize(svd.cols());
            else if (svd.m_computeThinV)
                m_workspace.resize(svd.rows());
            m_adjoint.resize(svd.cols(), svd.rows());
        }

        bool run(JacobiSVD<MatrixType, ColPivHouseholderQRPreconditioner>& svd, const MatrixType& matrix)
        {
            if (matrix.cols() > matrix.rows())
            {
                m_adjoint = matrix.adjoint();
                m_qr.compute(m_adjoint);

                svd.m_workMatrix = m_qr.matrixQR().block(0, 0, matrix.rows(), matrix.rows()).template triangularView<Upper>().adjoint();
                if (svd.m_computeFullV)
                    m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace);
                else if (svd.m_computeThinV)
                {
                    svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows());
                    m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace);
                }
                if (svd.computeU())
                    svd.m_matrixU = m_qr.colsPermutation();
                return true;
            }
            else
                return false;
        }

    private:
        typedef ColPivHouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
        QRType m_qr;
        TransposeTypeWithSameStorageOrder m_adjoint;
        typename internal::plain_row_type<MatrixType>::type m_workspace;
    };

    /*** preconditioner using HouseholderQR ***/

    template <typename MatrixType> class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreRowsThanCols, true>
    {
    public:
        void allocate(const JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd)
        {
            if (svd.rows() != m_qr.rows() || svd.cols() != m_qr.cols())
            {
                m_qr.~QRType();
                ::new (&m_qr) QRType(svd.rows(), svd.cols());
            }
            if (svd.m_computeFullU)
                m_workspace.resize(svd.rows());
            else if (svd.m_computeThinU)
                m_workspace.resize(svd.cols());
        }

        bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix)
        {
            if (matrix.rows() > matrix.cols())
            {
                m_qr.compute(matrix);
                svd.m_workMatrix = m_qr.matrixQR().block(0, 0, matrix.cols(), matrix.cols()).template triangularView<Upper>();
                if (svd.m_computeFullU)
                    m_qr.householderQ().evalTo(svd.m_matrixU, m_workspace);
                else if (svd.m_computeThinU)
                {
                    svd.m_matrixU.setIdentity(matrix.rows(), matrix.cols());
                    m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixU, m_workspace);
                }
                if (svd.computeV())
                    svd.m_matrixV.setIdentity(matrix.cols(), matrix.cols());
                return true;
            }
            return false;
        }

    private:
        typedef HouseholderQR<MatrixType> QRType;
        QRType m_qr;
        typename internal::plain_col_type<MatrixType>::type m_workspace;
    };

    template <typename MatrixType> class qr_preconditioner_impl<MatrixType, HouseholderQRPreconditioner, PreconditionIfMoreColsThanRows, true>
    {
    public:
        typedef typename MatrixType::Scalar Scalar;
        enum
        {
            RowsAtCompileTime = MatrixType::RowsAtCompileTime,
            ColsAtCompileTime = MatrixType::ColsAtCompileTime,
            MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
            MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
            Options = MatrixType::Options
        };

        typedef
            typename internal::make_proper_matrix_type<Scalar, ColsAtCompileTime, RowsAtCompileTime, Options, MaxColsAtCompileTime, MaxRowsAtCompileTime>::type
                TransposeTypeWithSameStorageOrder;

        void allocate(const JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd)
        {
            if (svd.cols() != m_qr.rows() || svd.rows() != m_qr.cols())
            {
                m_qr.~QRType();
                ::new (&m_qr) QRType(svd.cols(), svd.rows());
            }
            if (svd.m_computeFullV)
                m_workspace.resize(svd.cols());
            else if (svd.m_computeThinV)
                m_workspace.resize(svd.rows());
            m_adjoint.resize(svd.cols(), svd.rows());
        }

        bool run(JacobiSVD<MatrixType, HouseholderQRPreconditioner>& svd, const MatrixType& matrix)
        {
            if (matrix.cols() > matrix.rows())
            {
                m_adjoint = matrix.adjoint();
                m_qr.compute(m_adjoint);

                svd.m_workMatrix = m_qr.matrixQR().block(0, 0, matrix.rows(), matrix.rows()).template triangularView<Upper>().adjoint();
                if (svd.m_computeFullV)
                    m_qr.householderQ().evalTo(svd.m_matrixV, m_workspace);
                else if (svd.m_computeThinV)
                {
                    svd.m_matrixV.setIdentity(matrix.cols(), matrix.rows());
                    m_qr.householderQ().applyThisOnTheLeft(svd.m_matrixV, m_workspace);
                }
                if (svd.computeU())
                    svd.m_matrixU.setIdentity(matrix.rows(), matrix.rows());
                return true;
            }
            else
                return false;
        }

    private:
        typedef HouseholderQR<TransposeTypeWithSameStorageOrder> QRType;
        QRType m_qr;
        TransposeTypeWithSameStorageOrder m_adjoint;
        typename internal::plain_row_type<MatrixType>::type m_workspace;
    };

    /*** 2x2 SVD implementation
 ***
 *** JacobiSVD consists in performing a series of 2x2 SVD subproblems
 ***/

    template <typename MatrixType, int QRPreconditioner> struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, false>
    {
        typedef JacobiSVD<MatrixType, QRPreconditioner> SVD;
        typedef typename MatrixType::RealScalar RealScalar;
        static bool run(typename SVD::WorkMatrixType&, SVD&, Index, Index, RealScalar&) { return true; }
    };

    template <typename MatrixType, int QRPreconditioner> struct svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner, true>
    {
        typedef JacobiSVD<MatrixType, QRPreconditioner> SVD;
        typedef typename MatrixType::Scalar Scalar;
        typedef typename MatrixType::RealScalar RealScalar;
        static bool run(typename SVD::WorkMatrixType& work_matrix, SVD& svd, Index p, Index q, RealScalar& maxDiagEntry)
        {
            using std::abs;
            using std::sqrt;
            Scalar z;
            JacobiRotation<Scalar> rot;
            RealScalar n = sqrt(numext::abs2(work_matrix.coeff(p, p)) + numext::abs2(work_matrix.coeff(q, p)));

            const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)();
            const RealScalar precision = NumTraits<Scalar>::epsilon();

            if (n == 0)
            {
                // make sure first column is zero
                work_matrix.coeffRef(p, p) = work_matrix.coeffRef(q, p) = Scalar(0);

                if (abs(numext::imag(work_matrix.coeff(p, q))) > considerAsZero)
                {
                    // work_matrix.coeff(p,q) can be zero if work_matrix.coeff(q,p) is not zero but small enough to underflow when computing n
                    z = abs(work_matrix.coeff(p, q)) / work_matrix.coeff(p, q);
                    work_matrix.row(p) *= z;
                    if (svd.computeU())
                        svd.m_matrixU.col(p) *= conj(z);
                }
                if (abs(numext::imag(work_matrix.coeff(q, q))) > considerAsZero)
                {
                    z = abs(work_matrix.coeff(q, q)) / work_matrix.coeff(q, q);
                    work_matrix.row(q) *= z;
                    if (svd.computeU())
                        svd.m_matrixU.col(q) *= conj(z);
                }
                // otherwise the second row is already zero, so we have nothing to do.
            }
            else
            {
                rot.c() = conj(work_matrix.coeff(p, p)) / n;
                rot.s() = work_matrix.coeff(q, p) / n;
                work_matrix.applyOnTheLeft(p, q, rot);
                if (svd.computeU())
                    svd.m_matrixU.applyOnTheRight(p, q, rot.adjoint());
                if (abs(numext::imag(work_matrix.coeff(p, q))) > considerAsZero)
                {
                    z = abs(work_matrix.coeff(p, q)) / work_matrix.coeff(p, q);
                    work_matrix.col(q) *= z;
                    if (svd.computeV())
                        svd.m_matrixV.col(q) *= z;
                }
                if (abs(numext::imag(work_matrix.coeff(q, q))) > considerAsZero)
                {
                    z = abs(work_matrix.coeff(q, q)) / work_matrix.coeff(q, q);
                    work_matrix.row(q) *= z;
                    if (svd.computeU())
                        svd.m_matrixU.col(q) *= conj(z);
                }
            }

            // update largest diagonal entry
            maxDiagEntry = numext::maxi<RealScalar>(maxDiagEntry, numext::maxi<RealScalar>(abs(work_matrix.coeff(p, p)), abs(work_matrix.coeff(q, q))));
            // and check whether the 2x2 block is already diagonal
            RealScalar threshold = numext::maxi<RealScalar>(considerAsZero, precision * maxDiagEntry);
            return abs(work_matrix.coeff(p, q)) > threshold || abs(work_matrix.coeff(q, p)) > threshold;
        }
    };

    template <typename _MatrixType, int QRPreconditioner> struct traits<JacobiSVD<_MatrixType, QRPreconditioner>> : traits<_MatrixType>
    {
        typedef _MatrixType MatrixType;
    };

}  // end namespace internal

/** \ingroup SVD_Module
  *
  *
  * \class JacobiSVD
  *
  * \brief Two-sided Jacobi SVD decomposition of a rectangular matrix
  *
  * \tparam _MatrixType the type of the matrix of which we are computing the SVD decomposition
  * \tparam QRPreconditioner this optional parameter allows to specify the type of QR decomposition that will be used internally
  *                        for the R-SVD step for non-square matrices. See discussion of possible values below.
  *
  * SVD decomposition consists in decomposing any n-by-p matrix \a A as a product
  *   \f[ A = U S V^* \f]
  * where \a U is a n-by-n unitary, \a V is a p-by-p unitary, and \a S is a n-by-p real positive matrix which is zero outside of its main diagonal;
  * the diagonal entries of S are known as the \em singular \em values of \a A and the columns of \a U and \a V are known as the left
  * and right \em singular \em vectors of \a A respectively.
  *
  * Singular values are always sorted in decreasing order.
  *
  * This JacobiSVD decomposition computes only the singular values by default. If you want \a U or \a V, you need to ask for them explicitly.
  *
  * You can ask for only \em thin \a U or \a V to be computed, meaning the following. In case of a rectangular n-by-p matrix, letting \a m be the
  * smaller value among \a n and \a p, there are only \a m singular vectors; the remaining columns of \a U and \a V do not correspond to actual
  * singular vectors. Asking for \em thin \a U or \a V means asking for only their \a m first columns to be formed. So \a U is then a n-by-m matrix,
  * and \a V is then a p-by-m matrix. Notice that thin \a U and \a V are all you need for (least squares) solving.
  *
  * Here's an example demonstrating basic usage:
  * \include JacobiSVD_basic.cpp
  * Output: \verbinclude JacobiSVD_basic.out
  *
  * This JacobiSVD class is a two-sided Jacobi R-SVD decomposition, ensuring optimal reliability and accuracy. The downside is that it's slower than
  * bidiagonalizing SVD algorithms for large square matrices; however its complexity is still \f$ O(n^2p) \f$ where \a n is the smaller dimension and
  * \a p is the greater dimension, meaning that it is still of the same order of complexity as the faster bidiagonalizing R-SVD algorithms.
  * In particular, like any R-SVD, it takes advantage of non-squareness in that its complexity is only linear in the greater dimension.
  *
  * If the input matrix has inf or nan coefficients, the result of the computation is undefined, but the computation is guaranteed to
  * terminate in finite (and reasonable) time.
  *
  * The possible values for QRPreconditioner are:
  * \li ColPivHouseholderQRPreconditioner is the default. In practice it's very safe. It uses column-pivoting QR.
  * \li FullPivHouseholderQRPreconditioner, is the safest and slowest. It uses full-pivoting QR.
  *     Contrary to other QRs, it doesn't allow computing thin unitaries.
  * \li HouseholderQRPreconditioner is the fastest, and less safe and accurate than the pivoting variants. It uses non-pivoting QR.
  *     This is very similar in safety and accuracy to the bidiagonalization process used by bidiagonalizing SVD algorithms (since bidiagonalization
  *     is inherently non-pivoting). However the resulting SVD is still more reliable than bidiagonalizing SVDs because the Jacobi-based iterarive
  *     process is more reliable than the optimized bidiagonal SVD iterations.
  * \li NoQRPreconditioner allows not to use a QR preconditioner at all. This is useful if you know that you will only be computing
  *     JacobiSVD decompositions of square matrices. Non-square matrices require a QR preconditioner. Using this option will result in
  *     faster compilation and smaller executable code. It won't significantly speed up computation, since JacobiSVD is always checking
  *     if QR preconditioning is needed before applying it anyway.
  *
  * \sa MatrixBase::jacobiSvd()
  */
template <typename _MatrixType, int QRPreconditioner> class JacobiSVD : public SVDBase<JacobiSVD<_MatrixType, QRPreconditioner>>
{
    typedef SVDBase<JacobiSVD> Base;

public:
    typedef _MatrixType MatrixType;
    typedef typename MatrixType::Scalar Scalar;
    typedef typename NumTraits<typename MatrixType::Scalar>::Real RealScalar;
    enum
    {
        RowsAtCompileTime = MatrixType::RowsAtCompileTime,
        ColsAtCompileTime = MatrixType::ColsAtCompileTime,
        DiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_DYNAMIC(RowsAtCompileTime, ColsAtCompileTime),
        MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
        MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime,
        MaxDiagSizeAtCompileTime = EIGEN_SIZE_MIN_PREFER_FIXED(MaxRowsAtCompileTime, MaxColsAtCompileTime),
        MatrixOptions = MatrixType::Options
    };

    typedef typename Base::MatrixUType MatrixUType;
    typedef typename Base::MatrixVType MatrixVType;
    typedef typename Base::SingularValuesType SingularValuesType;

    typedef typename internal::plain_row_type<MatrixType>::type RowType;
    typedef typename internal::plain_col_type<MatrixType>::type ColType;
    typedef Matrix<Scalar, DiagSizeAtCompileTime, DiagSizeAtCompileTime, MatrixOptions, MaxDiagSizeAtCompileTime, MaxDiagSizeAtCompileTime> WorkMatrixType;

    /** \brief Default Constructor.
      *
      * The default constructor is useful in cases in which the user intends to
      * perform decompositions via JacobiSVD::compute(const MatrixType&).
      */
    JacobiSVD() {}

    /** \brief Default Constructor with memory preallocation
      *
      * Like the default constructor but with preallocation of the internal data
      * according to the specified problem size.
      * \sa JacobiSVD()
      */
    JacobiSVD(Index rows, Index cols, unsigned int computationOptions = 0) { allocate(rows, cols, computationOptions); }

    /** \brief Constructor performing the decomposition of given matrix.
     *
     * \param matrix the matrix to decompose
     * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
     *                           By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU,
     *                           #ComputeFullV, #ComputeThinV.
     *
     * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
     * available with the (non-default) FullPivHouseholderQR preconditioner.
     */
    explicit JacobiSVD(const MatrixType& matrix, unsigned int computationOptions = 0) { compute(matrix, computationOptions); }

    /** \brief Method performing the decomposition of given matrix using custom options.
     *
     * \param matrix the matrix to decompose
     * \param computationOptions optional parameter allowing to specify if you want full or thin U or V unitaries to be computed.
     *                           By default, none is computed. This is a bit-field, the possible bits are #ComputeFullU, #ComputeThinU,
     *                           #ComputeFullV, #ComputeThinV.
     *
     * Thin unitaries are only available if your matrix type has a Dynamic number of columns (for example MatrixXf). They also are not
     * available with the (non-default) FullPivHouseholderQR preconditioner.
     */
    JacobiSVD& compute(const MatrixType& matrix, unsigned int computationOptions);

    /** \brief Method performing the decomposition of given matrix using current options.
     *
     * \param matrix the matrix to decompose
     *
     * This method uses the current \a computationOptions, as already passed to the constructor or to compute(const MatrixType&, unsigned int).
     */
    JacobiSVD& compute(const MatrixType& matrix) { return compute(matrix, m_computationOptions); }

    using Base::cols;
    using Base::computeU;
    using Base::computeV;
    using Base::rank;
    using Base::rows;

private:
    void allocate(Index rows, Index cols, unsigned int computationOptions);

protected:
    using Base::m_cols;
    using Base::m_computationOptions;
    using Base::m_computeFullU;
    using Base::m_computeFullV;
    using Base::m_computeThinU;
    using Base::m_computeThinV;
    using Base::m_diagSize;
    using Base::m_info;
    using Base::m_isAllocated;
    using Base::m_isInitialized;
    using Base::m_matrixU;
    using Base::m_matrixV;
    using Base::m_nonzeroSingularValues;
    using Base::m_prescribedThreshold;
    using Base::m_rows;
    using Base::m_singularValues;
    using Base::m_usePrescribedThreshold;
    WorkMatrixType m_workMatrix;

    template <typename __MatrixType, int _QRPreconditioner, bool _IsComplex> friend struct internal::svd_precondition_2x2_block_to_be_real;
    template <typename __MatrixType, int _QRPreconditioner, int _Case, bool _DoAnything> friend struct internal::qr_preconditioner_impl;

    internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreColsThanRows> m_qr_precond_morecols;
    internal::qr_preconditioner_impl<MatrixType, QRPreconditioner, internal::PreconditionIfMoreRowsThanCols> m_qr_precond_morerows;
    MatrixType m_scaledMatrix;
};

template <typename MatrixType, int QRPreconditioner>
void JacobiSVD<MatrixType, QRPreconditioner>::allocate(Eigen::Index rows, Eigen::Index cols, unsigned int computationOptions)
{
    eigen_assert(rows >= 0 && cols >= 0);

    if (m_isAllocated && rows == m_rows && cols == m_cols && computationOptions == m_computationOptions)
    {
        return;
    }

    m_rows = rows;
    m_cols = cols;
    m_info = Success;
    m_isInitialized = false;
    m_isAllocated = true;
    m_computationOptions = computationOptions;
    m_computeFullU = (computationOptions & ComputeFullU) != 0;
    m_computeThinU = (computationOptions & ComputeThinU) != 0;
    m_computeFullV = (computationOptions & ComputeFullV) != 0;
    m_computeThinV = (computationOptions & ComputeThinV) != 0;
    eigen_assert(!(m_computeFullU && m_computeThinU) && "JacobiSVD: you can't ask for both full and thin U");
    eigen_assert(!(m_computeFullV && m_computeThinV) && "JacobiSVD: you can't ask for both full and thin V");
    eigen_assert(EIGEN_IMPLIES(m_computeThinU || m_computeThinV, MatrixType::ColsAtCompileTime == Dynamic) &&
                 "JacobiSVD: thin U and V are only available when your matrix has a dynamic number of columns.");
    if (QRPreconditioner == FullPivHouseholderQRPreconditioner)
    {
        eigen_assert(!(m_computeThinU || m_computeThinV) && "JacobiSVD: can't compute thin U or thin V with the FullPivHouseholderQR preconditioner. "
                                                            "Use the ColPivHouseholderQR preconditioner instead.");
    }
    m_diagSize = (std::min)(m_rows, m_cols);
    m_singularValues.resize(m_diagSize);
    if (RowsAtCompileTime == Dynamic)
        m_matrixU.resize(m_rows, m_computeFullU ? m_rows : m_computeThinU ? m_diagSize : 0);
    if (ColsAtCompileTime == Dynamic)
        m_matrixV.resize(m_cols, m_computeFullV ? m_cols : m_computeThinV ? m_diagSize : 0);
    m_workMatrix.resize(m_diagSize, m_diagSize);

    if (m_cols > m_rows)
        m_qr_precond_morecols.allocate(*this);
    if (m_rows > m_cols)
        m_qr_precond_morerows.allocate(*this);
    if (m_rows != m_cols)
        m_scaledMatrix.resize(rows, cols);
}

template <typename MatrixType, int QRPreconditioner>
JacobiSVD<MatrixType, QRPreconditioner>& JacobiSVD<MatrixType, QRPreconditioner>::compute(const MatrixType& matrix, unsigned int computationOptions)
{
    using std::abs;
    allocate(matrix.rows(), matrix.cols(), computationOptions);

    // currently we stop when we reach precision 2*epsilon as the last bit of precision can require an unreasonable number of iterations,
    // only worsening the precision of U and V as we accumulate more rotations
    const RealScalar precision = RealScalar(2) * NumTraits<Scalar>::epsilon();

    // limit for denormal numbers to be considered zero in order to avoid infinite loops (see bug 286)
    const RealScalar considerAsZero = (std::numeric_limits<RealScalar>::min)();

    // Scaling factor to reduce over/under-flows
    RealScalar scale = matrix.cwiseAbs().template maxCoeff<PropagateNaN>();
    if (!(numext::isfinite)(scale))
    {
        m_isInitialized = true;
        m_info = InvalidInput;
        return *this;
    }
    if (scale == RealScalar(0))
        scale = RealScalar(1);

    /*** step 1. The R-SVD step: we use a QR decomposition to reduce to the case of a square matrix */

    if (m_rows != m_cols)
    {
        m_scaledMatrix = matrix / scale;
        m_qr_precond_morecols.run(*this, m_scaledMatrix);
        m_qr_precond_morerows.run(*this, m_scaledMatrix);
    }
    else
    {
        m_workMatrix = matrix.block(0, 0, m_diagSize, m_diagSize) / scale;
        if (m_computeFullU)
            m_matrixU.setIdentity(m_rows, m_rows);
        if (m_computeThinU)
            m_matrixU.setIdentity(m_rows, m_diagSize);
        if (m_computeFullV)
            m_matrixV.setIdentity(m_cols, m_cols);
        if (m_computeThinV)
            m_matrixV.setIdentity(m_cols, m_diagSize);
    }

    /*** step 2. The main Jacobi SVD iteration. ***/
    RealScalar maxDiagEntry = m_workMatrix.cwiseAbs().diagonal().maxCoeff();

    bool finished = false;
    while (!finished)
    {
        finished = true;

        // do a sweep: for all index pairs (p,q), perform SVD of the corresponding 2x2 sub-matrix

        for (Index p = 1; p < m_diagSize; ++p)
        {
            for (Index q = 0; q < p; ++q)
            {
                // if this 2x2 sub-matrix is not diagonal already...
                // notice that this comparison will evaluate to false if any NaN is involved, ensuring that NaN's don't
                // keep us iterating forever. Similarly, small denormal numbers are considered zero.
                RealScalar threshold = numext::maxi<RealScalar>(considerAsZero, precision * maxDiagEntry);
                if (abs(m_workMatrix.coeff(p, q)) > threshold || abs(m_workMatrix.coeff(q, p)) > threshold)
                {
                    finished = false;
                    // perform SVD decomposition of 2x2 sub-matrix corresponding to indices p,q to make it diagonal
                    // the complex to real operation returns true if the updated 2x2 block is not already diagonal
                    if (internal::svd_precondition_2x2_block_to_be_real<MatrixType, QRPreconditioner>::run(m_workMatrix, *this, p, q, maxDiagEntry))
                    {
                        JacobiRotation<RealScalar> j_left, j_right;
                        internal::real_2x2_jacobi_svd(m_workMatrix, p, q, &j_left, &j_right);

                        // accumulate resulting Jacobi rotations
                        m_workMatrix.applyOnTheLeft(p, q, j_left);
                        if (computeU())
                            m_matrixU.applyOnTheRight(p, q, j_left.transpose());

                        m_workMatrix.applyOnTheRight(p, q, j_right);
                        if (computeV())
                            m_matrixV.applyOnTheRight(p, q, j_right);

                        // keep track of the largest diagonal coefficient
                        maxDiagEntry =
                            numext::maxi<RealScalar>(maxDiagEntry, numext::maxi<RealScalar>(abs(m_workMatrix.coeff(p, p)), abs(m_workMatrix.coeff(q, q))));
                    }
                }
            }
        }
    }

    /*** step 3. The work matrix is now diagonal, so ensure it's positive so its diagonal entries are the singular values ***/

    for (Index i = 0; i < m_diagSize; ++i)
    {
        // For a complex matrix, some diagonal coefficients might note have been
        // treated by svd_precondition_2x2_block_to_be_real, and the imaginary part
        // of some diagonal entry might not be null.
        if (NumTraits<Scalar>::IsComplex && abs(numext::imag(m_workMatrix.coeff(i, i))) > considerAsZero)
        {
            RealScalar a = abs(m_workMatrix.coeff(i, i));
            m_singularValues.coeffRef(i) = abs(a);
            if (computeU())
                m_matrixU.col(i) *= m_workMatrix.coeff(i, i) / a;
        }
        else
        {
            // m_workMatrix.coeff(i,i) is already real, no difficulty:
            RealScalar a = numext::real(m_workMatrix.coeff(i, i));
            m_singularValues.coeffRef(i) = abs(a);
            if (computeU() && (a < RealScalar(0)))
                m_matrixU.col(i) = -m_matrixU.col(i);
        }
    }

    m_singularValues *= scale;

    /*** step 4. Sort singular values in descending order and compute the number of nonzero singular values ***/

    m_nonzeroSingularValues = m_diagSize;
    for (Index i = 0; i < m_diagSize; i++)
    {
        Index pos;
        RealScalar maxRemainingSingularValue = m_singularValues.tail(m_diagSize - i).maxCoeff(&pos);
        if (maxRemainingSingularValue == RealScalar(0))
        {
            m_nonzeroSingularValues = i;
            break;
        }
        if (pos)
        {
            pos += i;
            std::swap(m_singularValues.coeffRef(i), m_singularValues.coeffRef(pos));
            if (computeU())
                m_matrixU.col(pos).swap(m_matrixU.col(i));
            if (computeV())
                m_matrixV.col(pos).swap(m_matrixV.col(i));
        }
    }

    m_isInitialized = true;
    return *this;
}

/** \svd_module
  *
  * \return the singular value decomposition of \c *this computed by two-sided
  * Jacobi transformations.
  *
  * \sa class JacobiSVD
  */
template <typename Derived> JacobiSVD<typename MatrixBase<Derived>::PlainObject> MatrixBase<Derived>::jacobiSvd(unsigned int computationOptions) const
{
    return JacobiSVD<PlainObject>(*this, computationOptions);
}

}  // end namespace Eigen

#endif  // EIGEN_JACOBISVD_H
